The adjoint representation inside the exterior algebra of a simple Lie algebra

For a simple complex Lie algebra g$\mathfrak{g}$ we study the space of invariants A=(?g^?⊗g^?)^g$A={(?{\mathfrak{g}}^{?}\otimes {\mathfrak{g}}^{?})}^{\mathfrak{g}}$, which describes the isotypic component of type g$\mathfrak{g}$ in ?g^?$?{\mathfrak{g}}^{?}$, as a module over the algebra of invariants (?g^?)^g${(?{\mathfrak{g}}^{?})}^{\mathfrak{g}}$. As main result we prove that A   is a free module, of rank twice the rank of g$\mathfrak{g}$, over the exterior algebra generated by all primitive invariants in (?g^?)^g${(?{\mathfrak{g}}^{?})}^{\mathfrak{g}}$, with the exception of the one of highest degree.     

A mathematical model of anorexia and bulimia

In this paper, we propose a mathematical model to study the dynamics of anorexic and bulimic populations. The model proposed takes into account, among other things, the effects of peers' influence, media influence, and education. We prove the existence of three possible equilibria that without media influences are disease‐free, bulimic‐endemic, and endemic. Neglecting media and education effects, we investigate the stability of such equilibria, and we prove that under the influence of media, only one of such equilibria persists and becomes a global attractor. Which of the three equilibria becomes global attractor depends on the other parameters. Copyright © 2014 John Wiley & Sons, Ltd.     

An operator splitting approach for the interaction between a fluid and a multilayered poroelastic structure

We develop a loosely coupled fluid‐structure interaction finite element solver based on the Lie operator splitting scheme. The scheme is applied to the interaction between an incompressible, viscous, Newtonian fluid, and a multilayered structure, which consists of a thin elastic layer and a thick poroelastic material. The thin layer is modeled using the linearly elastic Koiter membrane model, while the thick poroelastic layer is modeled as a Biot system. We prove a conditional stability of the scheme and derive error estimates. Theoretical results are supported with numerical examples. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1054–1100, 2015     

Tolerances as images of congruences in varieties defined by linear identities

An identity s = t is linear if each variable occurs at most once in each of the terms s and t. Let T be a tolerance relation of an algebra ${\mathcal{A}}$ in a variety defined by a set of linear identities. We prove that there exist an algebra ${\mathcal{B}}$ in the same variety and a congruence θ of ${\mathcal{B}}$ such that a homomorphism from ${\mathcal{B}}$ onto ${\mathcal{A}}$ maps θ onto T.     

A weak degeneracy revealing decomposition for the CANDECOMP/PARAFAC model

The CANDECOMP/PARAFAC (CP) model is a well known and frequently used tool for extracting substantial information from a three‐way data array. It has several useful characteristics and usually gives meaningful insights about the underlying structure of the data. However, in some cases it has a ‘strange’ behaviour suffering from the so‐called ‘degenerate solutions’, i.e. solutions where the components show a diverging pattern and are meaningless. Several authors have investigated the causes of degeneracy concluding that the phenomenon is due to a lack of minimum of the loss function. In this paper, we study the degeneracy of CP limiting our attention to the two‐component case. The study is done by introducing a canonical form, called 2DR, which is ‘weakly degeneracy revealing’. On the ground of this framework, degeneracy is studied along with some of the remedies proposed in the literature by using a Tucker3 model having a core in the 2DR form. The analysis gives new insights about the behaviour of the CP model and suggests new ideas on how to deal with degeneracy. Copyright © 2010 John Wiley & Sons, Ltd.     

The Baeyer–Villiger oxidation of ketones: A paradigm for the role of soft Lewis acidity in homogeneous catalysis

The Baeyer–Villiger oxidation of ketones to the corresponding esters or lactones is a valuable transformation that has been upgraded several times over the last century, from the original use of monopersulfuric acid as oxidant to more atom efficient and environmentally friendly oxidants such as hydrogen peroxide. The latter requires activation with organometallic complexes to explicate its oxidizing power. The catalytic version of the reaction can be achieved with several transition metal catalysts, but major differences are present among the various catalysts proposed in terms of scope of the reaction. In particular, most of the catalytic systems are active towards four-membered ring ketones leading to the corresponding substituted γ-butyro-lactones. Pt^II complexes characterized by the employment of chelating diphosphines turned out to be the most efficient in catalyzing the BV oxidation of a wider range of substrates, in particular cyclohexanones are suitable substrates and acyclic ketones can be converted into the corresponding esters, albeit with low turnover. As long as organometallic catalyzed BV reaction is concerned, Pt^II catalysts show the most versatile activity and selectivity. Such peculiar features are the result of the unique electronic properties of such metal combined with an easily tailored soft Lewis acid character modulated by the proper choice of the ancillary ligands. The enantioselective version of the reaction benefits from these properties and the compatibility of Pt^II species with water enabled the development of asymmetric catalytic BV reactions in water aided by the presence of micelles as dynamic self-assembled environments.     

Titanosilicate beads with hierarchical porosity: synthesis and application as epoxidation catalysts

Porous titanosilicate beads with a diameter of 0.5-1.5 mm (TiSil-HPB-60) were synthesized from a preformed titanosilicate solution with a porous anion-exchange resin as template. The bead format of this material enables its straightforward separation from the reaction mixture in its application as a liquid-phase heterogeneous catalyst. The material displays hierarchical porosity (micro/mesopores) and incipient TS-1 structure building units. The titanium species are predominantly located in tetrahedral framework positions. TiSil-HPB-60 is a highly active catalyst for the epoxidation of cyclohexene with t-butyl hydroperoxide (TBHP) and aqueous H(2)O(2). With both oxidants, TiSil-HPB-60 gave higher epoxide yields than Ti-MCM-41 and TS-1. The improved catalytic performance of TiSil-HPB-60 is mainly ascribed to the large mesopores favoring the diffusion of reagents and products to and from the titanium active sites. The epoxide yield and selectivity could be further improved by silylation of the titanosilicate beads. Importantly, TiSil-HPB-60 is a stable catalyst immune to titanium leaching, and can be easily recovered and reused in successive catalytic cycles without significant loss of activity. Moreover, TiSil-HPB-60 is active and selective in the epoxidation of a wide range of bulky alkenes.     

A Conjugated Thiophene‐Based Rotaxane: Synthesis,Spectroscopy, and Modeling

A dithiophene rotaxane 1 ?β‐CD and its shape‐persistent corresponding dumbbell 1 were synthesized and fully characterized. 2D NOESY experiments, supported by molecular dynamics calculations, revealed a very mobile macrocycle (β‐CD). Steady‐state and time‐resolved photoluminescence experiments in solution were employed to elucidate the excited‐state dynamics for both systems and to explore the effect of cyclodextrin encapsulation. The photoluminescence (PL) spectrum of 1 ?β‐CD was found to be blueshifted with respect to the dumbbell 1 (2.81 and 2.78 eV, respectively). Additionally, in contrast to previous observations, neither PL spectra nor the decay kinetics of both threaded and unthreaded systems showed changes upon increasing the concentration or changing the polarity of the solutions, thereby providing evidence for a lack of tendency toward aggregation of the unthreaded backbone.